Astronomers calculated the distance between the Earth and Sun in about 1769 when Venus passed across the sun. Please could you explain this calculation?

It goes like this: By 1769, Kepler's laws of planetary motion and Newton's law of gravity had been laid out and shown to work. The period of each planetary orbit had been measured, but not the absolute distances. Kepler's Third Law (which, really, is Newton's law of gravity written in a special form) relates the orbital period of each planet to its relative distance from the Sun. For instance, Kepler's Third Law tells us that if the orbit of Venus is 0.62 years (Earth years, that is), then its average distance from the Sun is 72% of the Earth-Sun distance. So astronomers knew the relative distances between each planet and the Sun, but they did not know how those distances compared to terrestrial units of length (like miles) or to the size of the Earth. Since the planets' orbital periods were all known, knowing a single absolute distance would give the distances to all other planets. Thus, if we knew the distance from the Earth to the Sun, then we'd know the size of Venus's orbit too, and the speed at which it moves. So all of these details can be related to one number: the Earth-Sun distance.

Imagine you and a friend are standing on one side of a street, but separated by a sizable distance. You friend is to your right, for concreteness. And both of you are staring at a single lamppost in front of you on the other side. A car approaches from your left. As you're staring at the lamppost, the car cuts through your line of sight first, then a short time later, it cuts through your friend's line of sight, right? Because your friend is looking at the lamppost from a different angle.

If you knew how far away you and your friend were standing, and the velocity of the car, and the time difference between you and your friend's crossing, you could use geometry to find your distance to the lamppost.

Now, move that analogy to the transit of Venus. You and your friend are at two separate observatories (at two far-apart locations on Earth), staring at the Sun, waiting for the transit. You will each see the transit happen at slightly different times. More importantly, you will each see Venus take a slightly different path across the Sun's surface, and you will measure slightly different durations for the transit. With those measurements, and some trigonometry, one can calculate the absolute distance to the Sun. In 1771, based on analysis of observations of the transits of Venus that occurred in 1761 and 1769, French astronomer Jérôme Lalande calculated a value of the astronomical unit that was just 2% higher than its actual (modern) value.

Here are some pages with more information on the transit observations (and some illustrations) and how they can be used to determine the Earth-Sun distance:

Note that there had been another fairly accurate calculation of the astronomical unit a century earlier, using the same principle (parallax) for observations of Mars. When Mars came close to Earth in 1672, simultaneous observations by Giovanni Cassini (in Paris) and Jean Richer (in French Guiana), comparing where Mars appeared relative to background stars, yielded a value of an astronomical unit that was about 7% higher than the modern value. This is discussed in more detail at:

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